Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit

Abstract

We investigate the relaxation problem and the diffusion phenomenon for the compressible Euler system with a time-dependent damping coefficient of the form $\frac{\hat{I}}{¼}(1+t{)}^{\hat{I}}»$ in ${\mathbb{R}}^d$ $(d\ge 1)$. We establish uniform regularity estimates with respect to the relaxation parameter $\epsilon$ and prove the global well-posedness of classical solutions to the Cauchy problem. In addition, we justify the global-in-time strong convergence of the solutions towards those of a general porous medium-type diffusion system, with an explicit rate of convergence, and for ill-prepared initial data. The core of our proof relies on a refined hypocoercivity framework combined with a new time-dependent frequency decomposition, both adapted to handle damping terms with time-dependent coefficients. This enables us to treat the overdamped regime $\hat{I}»\in (-\infty ,0)$ and the underdamped regime $\hat{I}»\in (0,1)$ for any $\hat{I}¼>0$, and also the borderline critical case $\hat{I}»=1$ under the improved condition $\hat{I}¼>2{\epsilon }^2$.